Particle Gibbs for non-linear models

using AdvancedPS
using Random
using Distributions
using Plots
using AbstractMCMC
using Random123
using Libtask

We consider the following stochastic volatility model:

\[ x_{t+1} = a x_t + v_t \quad v_{t} \sim \mathcal{N}(0, r^2)\]

\[ y_{t} = e_t \exp(\frac{1}{2}x_t) \quad v_{t} \sim \mathcal{N}(0, 1)\]

Here we assume the static parameters $\theta = (q^2, r^2)$ are known and we are only interested in sampling from the latent state $x_t$. We can reformulate the above in terms of transition and observation densities:

\[ x_{t+1} \sim f_{\theta}(x_{t+1}|x_t) = \mathcal{N}(a x_t, q^2)\]

\[ y_t \sim g_{\theta}(y_t|x_t) = \mathcal{N}(0, \exp(\frac{1}{2}x_t)^2)\]

with the initial distribution $f_0(x) = \mathcal{N}(0, q^2)$.

Parameters = @NamedTuple begin
    a::Float64
    q::Float64
    T::Int
end

mutable struct NonLinearTimeSeries <: AbstractMCMC.AbstractModel
    X::Array
    θ::Parameters
    NonLinearTimeSeries(θ::Parameters) = new(zeros(Float64, θ.T), θ)
end

f(model::NonLinearTimeSeries, state, t) = Normal(model.θ.a * state, model.θ.q)
g(model::NonLinearTimeSeries, state, t) = Normal(0, exp(0.5 * state)^2)
f₀(model::NonLinearTimeSeries) = Normal(0, model.θ.q)

Let's simulate some data

a = 0.9   # State Variance
q = 0.5   # Observation variance
Tₘ = 200  # Number of observation
Nₚ = 20   # Number of particles
Nₛ = 500  # Number of samples
seed = 1  # Reproduce everything

θ₀ = Parameters((a, q, Tₘ))
rng = Random.MersenneTwister(seed)

x = zeros(Tₘ)
y = zeros(Tₘ)

reference = NonLinearTimeSeries(θ₀)
x[1] = 0
for t in 1:Tₘ
    if t < Tₘ
        x[t + 1] = rand(rng, f(reference, x[t], t))
    end
    y[t] = rand(rng, g(reference, x[t], t))
end

Here are the latent and observation series:

plot(x; label="x", xlabel="t")

plot(y; label="y", xlabel="t")

Each model takes an AbstractRNG as input and generates the logpdf of the current transition:

function (model::NonLinearTimeSeries)(rng::Random.AbstractRNG)
    x₀ = rand(rng, f₀(model))
    model.X[1] = x₀
    score = logpdf(g(model, x₀, 1), y[1])
    Libtask.produce(score)

    for t in 2:(model.θ.T)
        state = rand(rng, f(model, model.X[t - 1], t - 1))
        model.X[t] = state
        score = logpdf(g(model, state, t), y[t])
        Libtask.produce(score)
    end
end

Here we use the particle gibbs kernel without adaptive resampling.

model = NonLinearTimeSeries(θ₀)
pgas = AdvancedPS.PG(Nₚ, 1.0)
chains = sample(rng, model, pgas, Nₛ; progress=false);

Each sampled trajectory holds a NonLinearTimeSeries model

particles = hcat([chain.trajectory.X for chain in chains]...) # Concat all sampled states
mean_trajectory = mean(particles; dims=2)

We can now plot all the generated traces. Beyond the last few timesteps all the trajectories collapse into one. Using the ancestor updating step can help with the degeneracy problem.

scatter(particles; label=false, opacity=0.01, color=:black, xlabel="t", ylabel="state")
plot!(x; color=:darkorange, label="Original Trajectory")
plot!(mean_trajectory; color=:dodgerblue, label="Mean trajectory", opacity=0.9)